Question: Divide the following complex numbers: $\dfrac{8 e^{\pi i}}{2 e^{7\pi i / 6}}$ (The dividend is plotted in blue and the divisor in plotted in green. Your current answer will be plotted orange.)
Solution: Dividing complex numbers in polar forms can be done by dividing the radii and subtracting the angles. The first number ( $8 e^{\pi i}$ ) has angle $\pi$ and radius 8. The second number ( $2 e^{7\pi i / 6}$ ) has angle $\frac{7}{6}\pi$ and radius 2. The radius of the result will be $\frac{8}{2}$ , which is 4. The difference of the angles is $\pi - \frac{7}{6}\pi = -\frac{1}{6}\pi$ The angle $-\frac{1}{6}\pi$ is negative. A complex number goes a full circle if its angle is increased by $2 \pi$ , so it goes back to itself. Because of that, angles of complex numbers are convenient to keep between $0$ and $2 \pi$ $-\frac{1}{6}\pi + 2 \pi = \frac{11}{6}\pi$ The radius of the result is $4$ and the angle of the result is $\frac{11}{6}\pi$.